Group #24908615 — subgroups

Subgroups — #24908615

Order 10624.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581784	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581717	C₂	1	2	yes	maximal
3	#19127607	C₂	1	1	yes	maximal, characteristic
4	#19127464	C₂	1	2	yes	maximal
5	#336×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₆₆	C₂²	1	1	yes	characteristic
7	#14581784	—	2	4	no	—
8	C₂×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
10	C₂×Dic₈₃⋊C₄	—	2	2	no	—
11	#14581717	—	2	4	no	—
12	C₈₃⋊#95	—	2	2	no	—
13	C₂²×C₈×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₆₆	—	2	2	no	—
16	C₂×Dic₁₆₆	—	2	2	no	—
17	C₂×Dic₁₆₆	—	2	1	no	—
18	C₂×Dic₁₆₆	Q₈	1	2	yes	—
19	C₂²×Dic₈₃	—	2	1	no	—
20	C₂²×Dic₈₃	—	2	2	no	—
21	C₂²×Dic₈₃	—	2	1	no	—
22	C₄⋊Dic₈₃	—	2	2	no	—
23	C₄⋊Dic₈₃	—	2	2	no	—
24	C₄⋊Dic₈₃	D₄	1	4	yes	—
25	Dic₈₃⋊C₄	—	2	8	no	—
26	C₂²×C₄×C₈₃	D₄	1	2	yes	—
27	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₈₃	—	2	2	no	—
29	C₂×C₈×C₈₃	—	2	2	no	—
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₄²×C₈₃	—	2	2	no	—
32	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₃	—	2	2	no	—
34	C₂×C₄×C₈₃	—	2	2	no	—
35	C₂×C₄×C₈₃	—	2	2	no	—
36	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
39	C₂×C₄×C₈₃	#41	1	2	yes	—
40	C₂×Dic₈₃	—	2	4	no	—
41	C₂×Dic₈₃	—	2	2	no	—
42	C₂×Dic₈₃	—	4	1	no	—
43	C₂×Dic₈₃	—	4	1	no	—
44	C₂×Dic₈₃	—	4	2	no	—
45	C₂×Dic₈₃	—	4	2	no	—
46	C₂×Dic₈₃	—	4	1	no	—
47	C₂×Dic₈₃	—	4	2	no	—
48	Dic₁₆₆	—	2	4	no	—
49	Dic₁₆₆	—	4	2	no	—
50	C₈×C₈₃	—	4	2	no	—
51	C₂²×C₈₃	#120	1	1	yes	characteristic
52	C₂²×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₈₃	#124	1	1	yes	characteristic
54	C₂²×C₈₃	#135	1	2	yes	—
55	C₂²×C₈₃	#137	1	2	yes	—
56	C₄×C₈₃	—	4	2	no	—
57	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₈₃	#124	1	2	yes	—
59	Dic₈₃	—	4	2	no	—
60	Dic₈₃	—	4	4	no	—
61	Dic₈₃	—	8	1	no	—
62	C₂×C₈₃	#393	1	1	yes	characteristic
63	C₂×C₈₃	#452	1	2	yes	—
64	C₂×C₈₃	#479	1	2	yes	—
65	C₂×C₈₃	#488	1	1	yes	characteristic
66	C₂×C₈₃	#491	1	1	yes	characteristic
67	#2026	—	83	1	no	maximal, Sylow
68	C₈₃	#2026	1	1	yes	characteristic, complemented, Sylow
69	C₂×#125	—	83	1	no	—
70	C₂×#103	—	83	2	no	—
71	#391	—	83	1	no	—
72	#340	—	83	2	no	—
73	#336	D₈₃	1	1	yes	characteristic
74	C₂²×Q₈	—	83	1	no	—
75	C₂²×C₈	C₂×D₈₃	1	2	yes	—
76	#125	—	166	4	no	—
77	C₂×C₄⋊C₄	—	83	2	no	—
78	C₂×C₄⋊C₄	—	83	1	no	—
79	C₂×C₄⋊C₄	—	166	2	no	—
80	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
81	#103	—	166	4	no	—
82	#95	—	166	2	no	—
83	C₂×Q₈	—	83	2	no	—
84	C₂×Q₈	—	166	2	no	—
85	C₂×Q₈	—	166	2	no	—
86	C₂×Q₈	—	166	1	no	—
87	C₂²×C₄	—	166	1	no	—
88	C₂²×C₄	—	166	1	no	—
89	C₂²×C₄	—	166	2	no	—
90	C₂²×C₄	D₃₃₂	1	2	yes	—
91	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	83	4	no	—
96	C₄⋊C₄	—	166	8	no	—
97	C₄⋊C₄	—	166	2	no	—
98	C₄⋊C₄	—	166	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
101	Q₈	—	332	2	no	—
102	Q₈	—	166	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	332	1	no	—
107	C₂×C₄	—	332	1	no	—
108	C₂×C₄	—	332	2	no	—
109	C₂×C₄	—	332	2	no	—
110	C₂×C₄	—	332	1	no	—
111	C₂×C₄	—	332	2	no	—
112	C₂×C₄	—	166	4	no	—
113	C₂×C₄	—	166	2	no	—
114	C₂×C₄	D₄×D₈₃	1	2	yes	—
115	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
116	C₂×C₄	#13050589	1	1	yes	characteristic
117	C₂×C₄	#13050590	1	2	yes	—
118	C₈	—	4	2	no	—
119	C₂²	#14581762	1	1	yes	characteristic
120	C₂²	#14581772	1	1	yes	characteristic
121	C₂²	#14581780	1	1	yes	characteristic
122	C₂²	#14581814	1	2	yes	—
123	C₂²	#14581825	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	332	4	no	—
126	C₄	—	332	2	no	—
127	C₄	—	664	1	no	—
128	C₄	#14581770	1	2	yes	—
129	C₄	#14581778	1	2	yes	—
130	C₂	#19127614	1	1	yes	characteristic
131	C₂	#19127865	1	2	yes	—
132	C₂	#19128009	1	2	yes	—
133	C₂	#19128056	1	1	yes	characteristic
134	C₂	#19128070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581784

C₂

yes

maximal, characteristic

C₂×#14581717

C₂

yes

maximal

#19127607

C₂

yes

maximal, characteristic

#19127464

C₂

yes

maximal

#336×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₂²

yes

characteristic

#14581784

—

C₂×C₄⋊Dic₈₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₃

C₂²

yes

—

C₂×Dic₈₃⋊C₄

—

#14581717

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

Q₈

yes

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

D₄

yes

—

Dic₈₃⋊C₄

—

C₂²×C₄×C₈₃

D₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₂²×C₈₃

#120

yes

characteristic

C₂²×C₈₃

C₂²⋊Q₈

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#137

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#124

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#393

yes

characteristic

C₂×C₈₃

#452

yes

—

C₂×C₈₃

#479

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₃

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₈₃

yes

—

#125

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

#103

—

166

—

#95

—

166

—

C₂×Q₈

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

—

166

—

C₂²×C₄

D₃₃₂

yes

—

C₂²×C₄

C₂²×D₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄⋊C₄

—

166

—

C₄²

—

100

C₂³

C₂×D₃₃₂

yes

characteristic

101

Q₈

—

332

—

102

Q₈

—

166

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

332

—

107

C₂×C₄

—

332

—

108

C₂×C₄

—

332

—

109

C₂×C₄

—

332

—

110

C₂×C₄

—

332

—

111

C₂×C₄

—

332

—

112

C₂×C₄

—

166

—

113

C₂×C₄

—

166

—

114

C₂×C₄

D₄×D₈₃

yes

—

115

C₂×C₄

Q₈×D₈₃

yes

characteristic

116

C₂×C₄

#13050589

yes

characteristic

117

C₂×C₄

#13050590

yes

—

118

C₈

—

119

C₂²

#14581762

yes

characteristic

120

C₂²

#14581772

yes

characteristic

121

C₂²

#14581780

yes

characteristic

122

C₂²

#14581814

yes

—

123

C₂²

#14581825

yes

—

124

C₄

—

125

C₄

—

332

—

126

C₄

—

332

—

127

C₄

—

664

—

128

C₄

#14581770

yes

—

129

C₄

#14581778

yes

—

130

C₂

#19127614

yes

characteristic

131

C₂

#19127865

yes

—

132

C₂

#19128009

yes

—

133

C₂

#19128056

yes

characteristic

134

C₂

#19128070

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₃₂
commutator, cosocle	C₂²×C₄×C₈₃	C₂³
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#336×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₄
hypercenter	#336	D₈₃
hypocenter	C₈₃	#2026
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₃₂

commutator, cosocle

C₂²×C₄×C₈₃

C₂³

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#336×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
68. C₈₃	#2026	semidirect

H ≅ G/N

type

68. C₈₃

#2026

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#14581784	2
C₂×#14581717	2
#19127607	2
#19127464	2
#336×C₈₃	2
#2026	83

subgroup

index

#336×C₈₃

Subgroups — #24908615

Conjugacy classes of subgroups

Complements

Maximal subgroups